SIGMOD Record, March 2009 (Vol. 38, No. 1)
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18
SIGMOD Record, March 2009 (Vol. 38, No. 1)
SIGMOD Record, March 2009 (Vol. 38, No. 1)
19
E, F. Codd ResearchDivision San Jose, California
REDUNDANCY AND CONSISTENCY OF RELATIONS STORED IN LARGE DATA BANKS
RJ 599(# 12343) August 19, 1969
ABSTRACT: The large, integrated data banks of the future will contain many relations of various degrees in stored form. It will not be unusual for this set of stored relations to be redundant. Two types of redundancy are defined and discussed. One type may be employed to improve accessibility of certain kinds of information which happen to be in great demand. When either type of redundancy exists, those responsible for control of the data bank should know about it and have some means of detecting any “logical” inconsistencies in the total set of stored relations. Consistency checking might be helpful in tracking down unauthorized (and possibly fraudulent) changes in the data bank contents.
DERIVABILITY,
20
SIGMOD Record, March 2009 (Vol. 38, No. 1)
Copies may be requestedfrom IBM ThomasJ. WatsonResearchCenter, PostOffice Box 218, Yorktown Heights, New York 10598
LIMITED DISTRIBUTION NOTICE - This report has been submitted for publicatiori elsewhere and has been issued as a Research Report for early dissemination of tts contents. As a courtesy t6 the intended publisher, it should not be widely distributed until after the dote of outside publication.
SIGMOD Record, March 2009 (Vol. 38, No. 1)
21
Some Linguistic Operations
2. 3.
Aspects
View of Data
Data Bank Control
6.
Redundancy
Derivability,
5.
Composition
3.4 Relations
and Consistency
Named and Stored
Join
3.3
Expressible,
Projection
3.2
4.
Permutation
3.1
on Relations
A Relational
1.
CONTENTS
19
11
10
9
6
5
5
5
4
1
22
SIGMOD Record, March 2009 (Vol. 38, No. 1)
SIGMOD Record, March 2009 (Vol. 38, No. 1)
23
model
a high
data
[l,
for
of
of the
information
evaluation
management
data
within
system.
is
the
is,
of
merits
of present
a clearer
is
has
derivation
of which
hand,
part
of
and consist-
forms
A a
maximal
a basis
second
it
other.
representations
the
that
and machine
yield
relative
permits
for
least
other
in the
limitations
of competing
that
redundancy,
view
the
and also
and logical
will
on the
on the
connections
relational
of
not
model,
discussed
view
for
only:
a means
graph
machine
to the
of
an explanamodel)
provides
(or
with
provides
one hand,
which
it
structure
It
respects
view
structure
of data
derivability, are
This
concerned
vogue.
on the
relational
systems,
the
standpoint)
a single
natural
in
language
confusions,
scope
Finally,
a logical
these
The network
treating
of the
relations.
(from
programs
is
several
Accordingly,
and organization
derivation
the
mistaking
paper.
a number
this
of relations,--
basis
advantage
its
in
any additional
retrieval
purposes.
with
paper of data.
21 presently
between
level
spawned
of
ency
sound
further
representation
independence
for
representation
view
of this
to be superior
superimposing
describing
without
of
network
appears
data
part
a relational
of
first
tion
The
INTRODUCTION
or
All
The ordering
(3) (4)
(5)
The
(2)
R
rows
corresponding
conveyed
domain.
by labeling
The significance
is
is
R
ordering
it
it
with
it
of
S 2,
the
partially
. ..)
part represents
sn of
- it
which
name of the
is
defined;
Sl,
significant
R;
properties:
of be remembered an essential
must
make use
2 binary,
degree
Rela-
of
domain
j th n.
element
of n-tuples,
distinct),
mathematical
second as the
its
a set
degree
An array
not
but
column
is
Sj
sl’
is
frequently
immaterial;
is
accepted
necessarily
its
to have
to
from
if
an n-tuple
of each
on which
domains
in
unary,
following
columns
the
of
distinct;
of
to the
are
we shall
expounded.
corresponds
rows
ordering
said
n n-ary.
called
is
refer
relations,
has the
being
sets
element
n
representation
row represents
Each
(1)
view
relation
relational
the
particular
of
reasons,
and degree
often
R
here
. . . . Sn (not
.I\‘e shall
first
above,
its
an n-ary
this
S2,
is used
of Data
on these
Sl,
1 are
representation
that
an array
expository
3 ternary,
As defined
degree For
has
sets
relation
View
and so on.
of degree
R.
S2,
of which
tions
of
from
each
a relation
is
R
term Given
The
A Relational
sense.
1.
1.
24
SIGMOD Record, March 2009 (Vol. 38, No. 1)
ship
3 7 1
A Relation
2
2
4
FIGURE 1:
subassembly)
of part
y.
is that
y)
(x,
component
part
is
respect
part.
component
(or
relation,
The
but
The meaning of
It is a binary
x is an immediate
called
domains),
to the relation.
identical
component.
with
(indicating
is called
two domains
depicted
relation
meanings
each of whose
distinct
possess
headings
may
of columns
by the name of
)
two columns
the ordering
2 shows,
why should
in Figure
domains,
As the example
have identical
matter?
the corresponding
are labeled
12
4
9
23
17
quantity
projects
of Degree 4
1
5
7
5
5
project
If the columns
3
1
ask:
2
part
to specified
of parts
of degree
in specified
a relation
the shipments-in-progress
1 illustrates
1
( supplier
One might
ship
quantities.
reflects
in Figure
suppliers
which
The example
from specified
called
2. 4
2:
deletion
part defined on the following
domains:
there might be an entity
considerat ion,
type relation.
an entity
such a relation
part)
call
type relation
called
In the example under
and we shall
of a
for a class The set of entities
description
type can be viewed as a relation,
w.
The individual (such as a particular The prototype
object
and suppliers.
of
informa-
ones, and alteration n-tuples.
of existing existing
is called an entity
[3].
for an individual an entity
given entity
of objects
is called
description
projects,
As time
may be subject to insertion
degrees.
of time-varying
Domains
for example, a data bank which contains tion about parts,
Consider,
Two Identical
7
7
are of assorted
each n-ary relation n-tuples,
with
6
4
6
6
2 3
5
5
5
part)
3
2
1
part
a data bank is a collection
These relations
that
A Relation
(
of components of any of its
additional
progresses,
relations.
We now assert
Figure
component
-
‘..
SIGMOD Record, March 2009 (Vol. 38, No. 1)
25
quantity
quantity
(5)
(6)
numbers
part
are.
that
(not
names.
the example,
if
number would
different
parts
key.
each entity.
were
This
is
it
always
in
such
given
distinct
be the case in
type may possess
is superfluous
would
part
a x.
is either
or a combination
if
while
called
each
of
to what
names, and
type part.
identify
be a key,
An entity
attributes
a combination)
more than one non-redundant
identifying
uniquely
is
is,
are present,
it
of attributes)
of the entity
(or combination)
which
part
colors
above correspond
weights,
part
(or combination
listed
part
all
A key is non-redundant
part
none of the participating
uniquely
that
not be.
attribute
would
a simple
color
above,
Such an attribute
entity.
type has values
one attribute
the attributes
entity
In the example
possible
The domains
all
a given
Normally,
are commonly called
is unlikely
it
at some instant,
that,
domains may be
While
of which
Each of these
some or all
as well.
conceivable
of values,
domains
on order
on hand
weight
in the data bank at any instant.
a pool
other
part
(4)
color
represented
in effect,
and possibly
part
(3)
name
part
(2)
number
part
(1)
or refer
two keys
exhibited
exhibited
type part,
The
which are
are inter-entity
we have discussed examples of relations
Both of these relations
to serving to identify
two keys referring
project.
relations.
such binary
of all
history
domain is the set
defined on the domain date and The salary
relation
eli(pIoyee is defined An element of the salary history
type relation
the domain salary.
domain is a binary
might be salary history.
domains on which the entity
be defined For example, one of the
may, in turn,
Thus, somedomains may have These relations
framework.
Non-atomic values can be discussed
on non-simple domains, and so on.
as elements.
the relational relations
within
(non-decomposable) values.
defined on simple domains - domains whose elements are atomic
So far,
relations.
component.
this. three keys, one
help clarify
part,
key
its
An
to distinct
an assembly containing
the first
in Figure 2 involves
refer
is that
relations.
entity
type serving distinct
either
1 involves
types supplier,
in Figure
a component, the second to identify
the commonentity
relation
for each of the entity
that
which
relation
inter-entity
inter-entity
called
in a data bank are between
to a commonentity
at least
of every
therefore,
relations
The examples in Figures 1 and 2 will
types
include
property
and are,
The relation
roles.
entity
domains
essential
types,
The remaining
3.
26
SIGMOD Record, March 2009 (Vol. 38, No. 1)
by
functions
calculus
normal in
form
of well-formed
needed can be defined
in prenex
the class
is in a precisely
expressions
request
which
H
[4].
to security
of the
and invoked
Any arith-
formulas
one-to-one
can be used
subject
specified
is
of data from the data
the
permanently,
R permits
less
H permits
together
*The second-order predicate calculus (rather than first-order) is needed because the domains on which relations are defined may themselves have relations as elements (see section 1 ) .
metic
predicate
with
specification
correspondence
in a set
The class
constraints.
of qualification
on such a retrieval
Action
perhaps
be as
R and the host
domains.
in storage.
indicate,
on those
of any subset
bank.
retrieval
for
relations
which
degrees
the declaration
by
would
paper to describe
command or problem
of domains,
features
sublanguage
salient
are represented
of various
R permits
specification
how these
H.
its
all
be a strong
for
Such a
sub-
modification)
power itself
syntactic
of this
(programming,
appropriate
and would
of linguistic
retrieval
as described
calculus.*
of data,
predicate
is not the purpose
the retrieval
declarations
relations
supporting
with
language
Let us denote
follows.
it
in detail,
While
(with
view of a universal
languages,
languages
embedding
retrieval
of host
such a language
oriented).
in a variety
for
proposed
candidate
other
a yardstick
would
language
provide
based on the second-order
the development
of a relational
Aspects
language
permits
The adoption
Some Linguistic
above,
2.
4.
between
often
be burdened
and sets.
directed
paths
For a relation
a relational
view
data
view,
users more relation
Two such paths
every
relation is stored, he using any combination remaining arguments (like Everest) (missing from many we shall call (logically)
as a nested
in which
n. n-ary
n, the number of
of a single
by the user
adopted
is factorial
*Once a user is aware that a certain will expect to be able to exploit it of its arguments as “knowns” and the as “unknowns,” because the information is there. This is a system feature current information systems), which symmetric exploitation of relations.
R.
depen-
names are associated relations.
since
and using
network
of degree
is
toward
in
deletion
relations.
is in the naming of
adopted
exploitation*
than with
necessary,
(n > 2) has to be expressed
if
it
the view
coining
symmetric
rather
with
if
are declared
With the usual
to be named and controlled
relation.
Again, relation
paths
binary
are needed to support
with
that
relations
used to retrieve
names than are absolutely
will
data elements
has on the language
effect
specified
the com-
machine
without
or sub-communities)
for
in their
from declared
user
are effective
by others,
elements
may be triggered
One important
dencies
which
may be present
relations
take
purposes
Insertions
for query
changes.
to declared
possible
to the individual
the form of removing
(as opposed
Some deletions
take
munity
that
Deletions
to any ordering
representation.
regard
for
may be fetched
new elements
may be held
A set so specified
or it
R.
the form of adding
only
in
SIGMOD Record, March 2009 (Vol. 38, No. 1)
27
for
with
if
these
the resulting
a permutation
relation
is applied
yields
with
two
is said
to the
the converse
operations.
systems
not
in
relations.
should,
Information
would
key role
for
and a ternary
Most users
representation
these
operations
) ) )
nota-
may not be a rela-
set
data bank control
two columns
has an array
relation,
More generally,
relation.
columns of an n-ary
Interchanging
relation
familiar
with
operations.
relations.
of their
are specifically
because
below
relation
the result
of the usual
of a binary
these
concerned
columns.
A binary
Permutation
be thoroughly
however,
3.1
and people
with
from other
concerned
relations
discussed
are introduced
designers
be directly
deriving
These operations
all
Nevertheless,
the union
is not a relation.
example,
The operations
relation
tion;
to them.
are sets,
on Relations
employ 7 names.
R ( project,
quantity
5 names in n-ary
in the form
entails
the 4-ary
n-ary
then 2n-1 names direct
For example,
n+l with
relations,
and, thus,
Q ( part,
notation
Since relations
Operations
binary
1.
1, which
be represented
of Figure
( supplier,
would
ship
are applicable
3.
in nested
P
notation,
relation
as described
in Section
of only
binary
tion
only
instead
involving
have to be coined
expression
R
except
relation
and
if
that
i
operator
i
th
k
.[:, i (n z k),
i2,
indiceb
in resulting
j rows
is the k-ary
Consider
R (j = 1, 2, . . . . k)
lIL(R)
is removed.
of
then
ik
column is column-~~,ii duplication
j
L = il,
of
relation
is a list
any desired
relation.
any
(striking
a relation
array
*
of the two operations.
is used to obtain
relaboth a
of that
is
of rela-
considerations
represents
1,
the ordering
of a relation
of the given
array
or combination
II
to be a projection
projection,
whose
columns
performance
for
in Figure
relation
to store
and then remove from the resulting
is an n-ary
L
of it,
unnecessary
in the TOWS. The final
A selection
is said
permutation, Thus,
advisable.
by a stored
by any permutation
certain
is logically
Suppose now we select
Projection
make it
duplication which
it
and some permutation
Although
leaves
ship
There are,
exploitation
which
symmetric
answerable
provides
of queries
which
permutation
to the set answerable
the set
out the others)
3.2
could
relation
tion.
identical
tions,
In a system
unchanged.
the identity
relation.
of the relation
of the given
4! = 24 permutations
we include
of columns
if
example,
to be a permutation
5.
28
SIGMOD Record, March 2009 (Vol. 38, No. 1)
2 4 2
5 1 7
has fewer
in Figure
)
relation
1
which have some
is derived.
relations,
from which it
case, the projection
of the Relation
1
supplier
of this
5
( project
A projection
Suppose we are given two binary
Join
1.
Projection
in this particular
A Permuted
than the relation
3:
II31 (ship)
3.
of Figure
in Figure
ship
of
A binary
a join of
R with
S.
If
R, S are binary
Any such ternary
relation
R is joinable
rela-
U
with
while Figure 5 shows
a ternary
relation
S if there exists
S.
loss of information,
of the
R, S, which
which preserves all
lI12(U) = R and II23(U) = S.
relation
R with
without
is called
such that
a binary
a join
are joinable
tion
relation
in the given relations?
to form a ternary
The example in Figure 4 shows two relations
information
relations
domain in common. Under what circumstances can we combine these
3.3
n-tuples
Note that,
Figure
is exhibited
the relation
6.
of
always exists
b, c):R(a,
R with
= R
It
if
is immediate that
(a, b) is a member of
1 2
2 2
Figure 4:
1
part 1
R ( supplier
with
1
2
1
join
join
is not the
project)
Relations
2
1
1
S (part
Two Joinable
)
join
Figure 6 shows another possible in Figure 4.
S.
However, this
the join shown in Figure 5 is the natural
R with
of the relations
R is joinable
b) A S(b, c))
I123(R*S) = S
lIl,(R*S)
for S(b, c).
S from Figure 4.
Note that
only one of
of
and
R and similarly
then
in such a case is the natural
I12(R) = II,(S),
S defined by
R*S = ((a,
R with
One join that
such that
where R(a, b) has the value true
join
S.
relations
SIGMOD Record, March 2009 (Vol. 38, No. 1)
29
1
2
2
S.
that
it is this
S.
*A function
R21(R)
or relation.
S is a function*,
no point
of
with respect to the joining
is a many-one binary
If either
R with
to
of
domain
element which gives rise
Such an element in the joining
It
under
is to be made)
an element (element 1)
(from Figure 4)
possesses more than one relative
is called a point of ambiguity
of joins.
R and also under
rJith the property
reveals
S
)
S (from Figure 4)
)
(the domain on which the join
of these relations
R with
2
2
Another Join of
1
1
2
1
2
project
1
part
1
U ( supplier
of the domain part
the plurality
2
1
2
R with
1
1
2
Join of
2
1
1
The Natural
Inspection
Figure 6:
Figure 5:
1
project
1
part
1
R*S ( supplier
R with
qualification
join would be an entirely
in the joining
of
P) * 3j(S(p,
relation
such that = R
nsl(U)
= T.
“23uJ) = s
“12W)
s))
R, S, T;
that
is,
with the following
a ternary
or
S can sometimes be
R, gZ1(R),
separate
R and S, a relation
s) A R(s, p)),
j) A T(j,
s) * 3p(R(s, P) A S(p, j))
S(P, j) * 3s(T(j,
Rb,
T(j,
then we may form a three-way join of
(5)
(4)
(3)
f12(T) = n,(R)
(2)
= n,(s)
n,(n
(1)
:
and supplier
S”
R (as well
R with
Suppose we are given,
R with
from sources independent of
on the domains project
properties
T
with
“of
R with
In such a case,
In Figure 4, none of the relations
S), and this
resolved by means of other relations. can derive
S.
S is the only join of
R with
because S might be joinable
is a function.
Ambiguity
S, $1(S)
consideration.
as
R with
Note that the reiterated
join of
can occur in joining
is necessary,
s .
the natural
ambiguity
7.
30
SIGMOD Record, March 2009 (Vol. 38, No. 1)
under
Figure
2
Binary
a
b
2
7:
a
FL)
S(p
Relations
y
x),
e
b with
d
e
d
i)
b
a
a
y=d;
T
a Plurality
2=2
x
of
and
e
d
d
2
2
1
~1
of
J-Joins
x
T with
to join-
a relative
of Cyclic
T(L
7 the points
T, and
than
under
to
the relations respect
(say y),
with
must be a relative
under
y
S with
of ambiguity
in Figure
of
x = a;
Note that
property.
R.
a relative
1
R (2
(say point
21, and, furthermore,
S, z
(say
S
points
To be specific,
constraints
the circumstances
much more severe
an example),
of 2-joins.
entail
7, 8 for
J-join
r’elation
to distinguish
more than one cyclic
= T.
“34(V) for
=s
3-join
be a quaternary
a cyclic would
33(v)
must possess
a plurality
R with
have this
z
under
R
ing
R, S, T
for
those
can occur
which
be called
“12W) = R
3-join
will
is possible
this
it
which
While
such that
(see Figures
V
from a --linear
Such a join
exist
it
8.
of degree
a2,
Thus,
n-l
y(R*S*T).
the natural
R
a b b
2 2 2
cyclic
a2,
the cyclic
. . . . an-l,
of
by tying
R, S, T
by
an) A al = an).
relation
n
a
because
produces of degree
which
3-join
d))
7
R, S,
in Figure
hand side To obtain y
e
d
e
d
d
relations
c) A T(c,
is an n-ary
from a relation
the operator
if
a
2
binary
b) A S(b,
of three
is associative.
. . . . a,,l):R(al,
may now represent
= ((al,
the expression
We
y(R)
(*)
a
1
E i>
of the Relations
U’(s
are not needed on the left
we introduce
ends together.
relation
counterpart,
J-join
3-Joins
e
d
e
d
b, c, d):R(a,
linear
Two Cyclic
2-join
parentheses
the natural
its
by
R*S*T = {(a,
is given
where
T
8:
The natural
Figure
b
2 b
a
2 2
a
E i)
1
u (2
,
I
..
SIGMOD Record, March 2009 (Vol. 38, No. 1)
31
counterparts
We
tion
3.4
applied
were
to functions.
is probably We shall
familiar
as if
discuss
a
a generalization
of
of composi-
of
of linear
were
n-joins
the notion
this
of
on the
A similar
it
relation
and cyclic
with
Take the
product
domains
so.
not so,
S and call
B.
The notions
S
applicable.
C.
the linear
degrees.
with
are now directly
B,
of
a binary
s-p domains
this
these
of
R
R, and
were
of
p
to make it r-p
this
on suppose
Take the Cartesian
we can treat
it
on the domains
Similarly,
R as if
of assorted
The reader
of
of the last
domains
A.
If
of the first
R, and call
product
S.
domains
permutations
domains of
r
For simplicity, of the
appropriate
s
p
p < s).
new domain
can be taken
J-join
Composition
n relations
approach
A, B.
can treat
relation
and cyclic
binary
domains
C.
Cartesian
p
this
product
R, and call
of the last
apply
of the
the Cartesian
always
we could
Now, take
(p < r,
are the last
p
domains
domains
the first
p
their
are to be joined
the case of two relations
s) which
Consider
are not
may be appropriate, which
and
relations
3-join
of n binary
and cyclic
of relations
A few words
the joining
(degree
S
(degree
r),
binary.
necessarily
regarding
of linear
to the joining
of the notions
n 2 3) is obvious.
natural
however,
(where
their
Extension
if
and only
S such that if
they
Figure
9:
in Figure
is exhibited
Thus,
there
of
R with
R-S
The Natural
6).
4, their
2 1 2
1 2 2
R with
A
of
supplier
)
from the join
S
S. is
S
(from
exhibited
the
Figure
composition
/’
R
the exist-
natural
by
R with
R with
S defined
of
of
1
Composition
of
are com-
U
is a
given
S does not imply
However,
9 and another
Figure in Figure 10 (derived
/
( project
in Figure
is exhibited:
R, Si from
R-5 =,,.h13(R*S).
of : R with
join
T
a join
R, S.
two relations
exists
Our
are based very
relations.
and joinability
are joinable.
to the natural
composition
the relations
composition
Taking
the natural
S if
two relations
of join
of more than one composition
Corresponding
existence
to binary
and composability
first
T = II13(U).
R with
ence of more than one join
posable
with
of
Suppose we are given composition
above.
directly
it
of composition
and apply
on the definitions
concept
definitions
that
9.
4)
32
SIGMOD Record, March 2009 (Vol. 38, No. 1)
10:
Another
joins
c
several
of point
4)
com-
Figure
d
e
2
2
binary
what
of relations
relations
are
)
need to be actually
operations
to
degrees)
which
g f
g
f
g f
project
joining
to make use of these
of pairwise
may be of different
as extension
now proceed
in considering
We
e
d
C
of
a, b, d, e.
Only One Composition
to pairs
(and which
of composition
Many Joins,
C
2
11:
C
1 C
b
b
(part
have
the ambiguity
which
S, because
1
s
the points
R with
a
part)
made via
in composing
the same pattern
such relations.
stored.
lost
Note that
of two relations
one composition.
an example
a
Figure
Extension
on relations
(from
the number of distinct
S
1
( supplier
not necessarily
follows
is
but only
11 shows
associations
Figure
R
exist,
R with
)
may be as few as one or as many as the number of distinct
joins .
unambiguous
of
1
2
Composition
2
supplier
1
T ( project
When two or more joins
positions
Figure
10.
Associated
Expressible,
the stored
(3)
operators
of the predicate
relational quantifiers
set
The stored
This
included
set,
should in the named set.
then it
should be given
such a relation
on some unnamed but expressible that
whose
a public
be
relation
it
set would
relations
and we assume that
in the data bank.
of all
set
-
in the
and the
by means of simple
relations
connectives
of the expressible
of the named set,
stored
is the collection
in the stored name and thereby
is a subset
can identify
subset.
set
to such proportions included
grows
the traffic
is.
If
be a subset
are actually
small
This a very
names.
the user
normally
values
usually
public
data bank which
is the collection
calculus.
Such names of relations,
of all
from simple
of data to be retrieved.
for
which
of
language
of relations
collections
in the retrieval
such as =, logical
are constructed
expressions
sets
of defining
The named set
set
is the collection
by expressions
set
the named set
(2) set
the expressible
(1)
the purpose
can be designated
Relations
a data bank are three
Named and Stored with
The expressible
relations:
4.
.
SIGMOD Record, March 2009 (Vol. 38, No. 1)
33
and tie
mainly
regarding
R
is
any time
virtually
(for
relations
named
from
the
yields
yields
from
stored
of
we must
value
for
of R
S.
set
take
the
at times
This
natural
relations
that
loads,
exclude
projections, members
a set
S
and changes
interaction
stored
hand,
membership
in
of
named
in the list of operations, a join and a projection.
set,
a correct
R
of permutations,
derivable
and
other
of past
with
the
language
by
community
in the
investment
the
in the
of
On the belong
users,
set.
natural
together
retrieval
belong
the
and Consistency
of the
*We can omit natural composition because it is a combination of
of operations
which
needs
of
in
of time)
and not
Such definitions
ever-increasing
logical
set,
composition,
by name as a result
Redundancy
factors.
a sequence*
and ties
exists
A relation
Derivability,
in these
sequence
joins,
there
5.
place
on the
scope
y).
the
relations
transaction
the
which
in
requirements
mainly
are based
performance
regarding
decisions
relations
of these
relations
the
0, *,
on the
on the
in
set
(independent
named
stored
in the
natural
which
(II,
be within
and particularly
based
using
are
Decisions
must
are
by expressions
which
of relations
defined
operators
programs
users,
set
names
are
relations
permutation-projection,
set
expressions
join
the
involving
stored
Those
if
R.
Note
because
actually
strong
set,
is
base
the
on the
resulting
other
in both
or data
weak
of users.
redundancies
changes
removable
are
from
sets.
the
at
acceptable.
stored
Strong the
of
removable
in
natural
appear
not
at
of other
time. inherent
they
of
members
be the
They are
are
and stored
performance
are
named hand,
the
might
If
set
bank
of queries
and
data
some join
set storage
contains other
it
one at some other
question
administrator.
community
speaking,
in
if from of
the
load
of the to be stored
rest
R
there
of
contains
to save
the
stored
with
it
likely
the
if
insertions,
in the
derivable
values
specified,
a heavy
redundant
a projection
not
with
updates,
in order
is
from
convenience,
relations
interaction
weakly
The join
times
is
and an unnatural
set.
of
be justified
kinds
an environment
at all
of the
appear
providing
redundancies,
they
system
by the all,
needs
logical
Generally
some time
one at
is of the
but
user
is
redundant
derivable of
to perform
which
of relations
one relation set,
other
in
redundancy
members
the
least
Only
as time
to the
as well
A set
relations.
would
relative
-deletions.
space
for
be non-strongly-redundant
often
sense
will
this
in
is
join
made to the
to take.
strongly
named set
which
is
being natural join
redundant
the
one relation
of relations
as to which
that,
are
While
least
A set
no question
S).
changes
members.
at
is
and
at which
11.
34
SIGMOD Record, March 2009 (Vol. 38, No. 1)
three relations
most probably
*A binary relation a function.
will
is complex if neither
about each named relation,
system lacks - and it
of time between the member relations.
information
of statements
sense,
it
semantic
it nor its
converse is
cannot deduce the -
- detailed
If the information
of the redundancies which hold independent
associate with that set a collection
which define all
we shall
is redundant in either
possesses
join
do exist
of some cyclic
However, constraints
Thus, this set of relations
Whenever a set of relations
a weak redundancy.
of the three of them.
from time to time in
with the possi-
Hence, none of them is
between them, since each is a projection
from the other two.
of any two.
of ambiguity occurring
joining
of points
are complex* relations
part by supplier
s
is supplied at least one kind of
j
project
All
derivable
p to at least one
j
s)
the potential
bility
supplies part
R, S, T
p is supplied by at least one supplier
s
relations
consider the case
to project
part
project
supplier
T(j,
S(P, j)
R(s, P)
with meanings as follows:
in which there are binary
As an example of a weak redundancy,
cited previously
12.
to the named set.
It might,
over
this
constraint.
= gl(R)
such that
there is an element
an instantaneous
large and highly
of relations, variable.
snapshot of a collection some of which may be very
taking
problems (which we shall not discuss here) in
II12(R) and (b, c) is in lI12(S).
II12(T) (a, b) is in
(a, c) in the relation b
R, S, T in the system) and
two columns of each of
for every element pair
HZ(T) = h,(S)
Rl(T)
There are practical
(3)
(2)
(1)
determine whether
R, for making this
the values stored for An algorithm
(in whatever way they are represented
check would examine the first
satisfy
statement
is a composition of IIl2(R) with l’I12(S)tt,
we may check from time to time that S, T
C consistent
For example, given stored relations
with the constraint
redundanci’es .
call
and an associated
as it does or does not comply with
we shall
C of relations
statements, according
together
“IIk2(T)
R, S, T
the stated
or inconsistent
set of constraint
Given a collection
but such attempts would be fallible.
a period of time, make attempts to -induce the redundancies,
redundancies applicable
‘\
SIGMOD Record, March 2009 (Vol. 38, No. 1)
35
Bank
S (part,
him that
wishes
different
Laboratory
for
helpful
E. B.
Altman
of rela-
discussions.
Dr. F. P. Palermo and Dr.
ACKNOWLEDGMENT
subcollections
by
of system
Ideally,
now need to be
selections
by
if
the system
the system
interval
so that
of an incon-
and deletions
in the collection.
data bank.
to thank
of the San Jose Research
The author
tions
for
R, T
time
Alternatively, insertions
to make different
consistency
to inconsistency
be possible
reasonable
internally,
such and such relations
in an individual
should
reaction
it
changed to restore
informing
officer.
in making
assist
could
the user
the security
notify
some
be logged
The generation
in the relations
within
say (2, 5) in the relation who supplies
due
input
may arise
An example of inadequate
of relations
2 has no supplier
section).
could
could
not remedied
kind
insertions
were
of this
5 (see previous
when part
appropriate
it
sistency
input.
in a collection
of a new element,
or faulty
project)
is the insertion
project
Control
Inconsistencies
Data
to inadequate
6.
4.
3.
2.
1.
Random
A. Church, Princeton,
“An Introduction 1956.
Access
to Mathematical
Logic
I,”
Fall
Annual Review Pergamon
Processing,”
Proceedings
File Processing,” 5, 13, pp. 77-149,
for
G. H. Mealy, “Another Look at Data,” Joint Computer Conference, 1967.
C. McGee, “Generalized in Automatic Programming Press, 1969.
W.
C. W. Bachman, YSoftware Datamation, April 1965.
REFERENCES
13.
36
SIGMOD Record, March 2009 (Vol. 38, No. 1)
